A) \[\frac{55}{336}\]
B) \[\frac{17}{105}\]
C) \[\frac{1}{6}\]
D) \[\frac{19}{112}\]
Correct Answer: A
Solution :
\[\sum\limits_{n=1}^{5}{\frac{1}{n\left( n+1 \right)\left( n+2 \right)\left( n+3 \right)}}=\frac{k}{3}\] \[\Rightarrow \]\[\frac{1}{1.2.3.4}+\frac{1}{2.3.4.5}+.........+\frac{1}{6.6.7.8}=\frac{k}{3}\] \[\Rightarrow \]\[\frac{1}{3}\left[ \frac{1}{1.2.3}-\frac{1}{6.7.8} \right]=\frac{k}{3}\]\[\Rightarrow \]\[\frac{1}{3}\left[ \frac{1}{6}-\frac{1}{336} \right]=\frac{k}{3}\] \[\Rightarrow \]\[k=\frac{55}{336}\]
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